Extensions of diffeomorphisms from $R^3$ to $S^3$. Is there a convenient theorem about which diffeomorphisms $f: \mathbb R^3\rightarrow\mathbb  R^3$ can be extended to diffeomorphisms $\overline{f}: S^3\rightarrow S^3$?
That is, given a diffeomorphism $f:\mathbb R^3\rightarrow\mathbb R^3$, when does there exist an inclusion $i:\mathbb R^3\rightarrow S^3$ inducing a homeomorphism between $R^3$ and $i(\mathbb R^3)$ such that the map $\overline{f}: S^3\rightarrow S^3$ which fixes the point at infinity and is equal to $i\circ f\circ i^{-1}(x)$ for every other $x\in S^3$ is a diffeomorphism?
 A: Here's one approach for stereographic projection, which can be generalized to any inclusion which maps onto the sphere minus one point.
Let $i:\mathbb{R}^3\to S^3$ be the standard stereographic projection, and $p\in S^3$ such that $i(\mathbb{R}^3)=S^3\setminus p$.
The extension $\bar{f}$ is automatically determined on $S^3\setminus p$, and if a smooth extension exists, then the limits at $p$ of $\bar{f}$ and all of its derivatives must be well behaved. This corresponds to a set of conditions "at infinity" for $f$ and its derivatives. A simple way to keep track of these is by choosing a convenient chart around $p$.
For instance, in this case we can use the reversed stereographic projection as a chart. The local representative $g:\mathbb{R}^3\setminus 0\to\mathbb{R}^3\setminus 0$ is given by
$$
g(x)=f\left(\frac{x}{\|x\|^2}\right)
$$
and $f$ has a smooth extension iff $g$ and all of its partial derivatives have finite limits at zero. Furthermore, this extension is a diffeomorphism if the limit of $dg$ at zero is full rank, i.e. if the limit of the Jacobian determinant of $g$ at zero is nonzero.
